Problems

A natural number \(p\) is called prime if the only natural divisors of \(p\) are \(1\) and \(p\). Prime numbers are building blocks of all the natural numbers in the sense of the The Fundamental Theorem of Arithmetic: for a positive integer \(n\) there exists a unique prime factorization (or prime decomposition) \[n = p_1^{a_1}p_2^{a_2}...p_r^{a_r}.\] Today we will explore how unusual prime numbers are.
Essentially there is only one way to write an integer number as a product of prime numbers, where some of the prime numbers in the product can appear multiple times.

One can hardly imagine modern life without numbers, but have you wondered when and how the numbers were invented? It turns out people started using numbers about \(42000\) years BCE supposedly to mark the dates in calendar. But how do we represent the numbers in writing? Well, there are two ways: examples of the first abstract numeral systems are generally tallying systems, the ones where the value or contribution of a digit does not depend on its position, a good example is the famous Roman numeral system: \(I\, V\, X\, L\, C\, D\, M\), here a digit has only one value: \(I\) means one, \(X\) means ten and \(C\) a hundred. However, one might struggle to express large numbers in Roman system.
Majority of ancient civilisations, Sumerian, Egyptian, Babylonian, Chinese, Japanese, Indian used what is called positional numeral systems, where the contribution of a digit to the value of a number is the value of the digit multiplied by a factor determined by the position of the digit. All these systems, even when invented independently, have something in common, they are what is called "base-\(10\)".
Try to guess why do we use the decimal numeral system, which has exactly \(10\) digits in our everyday use. Because it does not actually have to be \(10\) digits, it could easily be \(3,8,16\), the binary system (with only digits \(0\) and \(1\)) is used in all electronic devices, since it is enough to represent any bit of information we might possibly know.

There exist various ways to prove mathematical statements, one of the possible methods, which might come handy in certain situations is called Proof by contradiction. To prove a statement we first assume that the statement is false and then deduce something that contradicts either the condition, or the assumption itself, or just common sense. Thereby concluding that the first assumption must have been wrong, so the statement is actually true.

In a lot of geometric problems the main idea is to find congruent figures. We call two polygons congruent if all their corresponding sides and angles are equal. Triangles are the easiest sort of polygons to deal with. Assume we are given two triangles \(ABC\) and \(A_1B_1C_1\) and we need to check whether they are congruent or not, some rules that help are:

• If all three corresponding sides of the triangles are equal, then the triangles are congruent.

• If, in the given triangles \(ABC\) and \(A_1B_1C_1\), two corresponding sides \(AB=A_1B_1\), \(AC=A_1C_1\) and the angles between them \(\angle BAC = \angle B_1A_1C_1\) are equal, then the triangles are congruent.

• If the sides \(AB=A_1B_1\) and pairs of the corresponding angles next to them \(\angle CAB = \angle C_1A_1B_1\) and \(\angle CBA = \angle C_1B_1A_1\) are equal, then the triangles are congruent.

At a previous geometry lesson we have derived these rules from the axioms of Euclidean geometry, so now we can just use them.

Today we will be solving problems using the pigeonhole principle. What is it? Simply put, we are asked to place pigeons in pigeonholes, but the number of pigeons is larger than the number of pigeonholes. No matter how we try to do that, at least one pigeonhole will have to contain at least 2 pigeons. By ”pigeonholes” we can mean any containers and by ”pigeons” we mean any items, which are placed in these containers. This is a simple observation, but it is helpful in solving some very difficult problems. Some of these problems might seem obvious or intuitively true. Pigeonhole principle is a useful way of formalising things that seem intuitive but can be difficult to describe mathematically.

There is also a more general version of the pigeonhole principle, where the number of pigeons is more than \(k\) times larger than the number of pigeonholes. Then, by the same logic, there will be one pigeonhole containing \(k+1\) pigeons or more.

A formal way to prove the pigeonhole principle is by contradiction - imagine what would happen if each pigeonhole contained only one pigeon? Well, the total number of pigeons could not be larger than the number of pigeonholes! What if each pigeonhole had \(k\) pigeons or fewer? The total number of pigeons could be \(k\) times larger than the number of pigeonholes, but not greater than that.

Draw how Robinson Crusoe should put pegs and ropes to tie his goat in order for the goat to graze grass in the shape of a parallelogram.

Draw a picture how Robinson used to tie the goat and the wolf in order for the goat to graze the grass in the shape of half a circle.

Draw a picture how Robinson used to tie the goat and the wolf in order for the goat to graze the grass in the shape of a young moon (see the picture below)

Draw a picture how Robinson used to tie the goat and the wolf in order for the goat to graze the grass in the shape of half a ring.

The dragon locked six dwarves in the cave and said, "I have seven caps of the seven colors of the rainbow. Tomorrow morning I will blindfold you and put a cap on each of you, and hide one cap. Then I’ll take off the blindfolds, and you can see the caps on the heads of others, but not your own and I won’t let you talk any more. After that, everyone will secretly tell me the color of the hidden cap. If at least three of you guess right, I’ll let you all go. If less than three guess correctly, I’ll eat you all for lunch." How can dwarves agree in advance to act in order to be saved?